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THE TRUNC ELEMENT IN ANY DIMENSION AND APPLICATION TO A MODIFIED POISSON EQUATION

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arXiv 2024年

作者： Li, Hongliang Ming, Pingbing Zhou, Yinghong Department of Mathematics Sichuan Normal University Chengdu610066 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences No. 55 East Road Zhong-Guan-Cun Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We introduce a novel TRUNC finite element in n dimensions, encompassing the traditional TRUNC triangle as a particular instance. By establishing the weak continuity identity, we identify it as crucial for error estimate. This element is utilized to approximate a modified Poisson equation defined on a convex polytope, originating from the nonlocal electrostatics model. We have substantiated a uniform error estimate and conducted numerical tests on both the smooth solution and the solution with a sharp boundary layer, which align with the theoretical predictions. Copyright © 2024, The Authors. All rights reserved.

关键词： Electrostatics

Simultaneous dose and dose rate optimization (SDDRO) of the FLASH effect for pencil-beam-scanning proton therapy

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Simultaneous dose and dose rate optimization (SDDRO) of the ...

作者： Gao, Hao Liu, Jiulong Lin, Yuting Gan, Gregory N Pratx, Guillem Wang, Fen Langen, Katja Bradley, Jeffrey D Rotondo, Ronny L Li, Harold H Chen, Ronald C Department of Radiation Oncology University of Kansas Medical Center Kansas CityKS United States LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Department of Radiation Oncology Stanford University StanfordCA United States Department of Radiation Oncology Emory University AtlantaGA United States

Purpose: Compared to CONV-RT (with conventional dose rate), FLASH-RT (with ultra-high dose rate) can provide biological dose sparing for organs-at-risk (OARs) via the so-called FLASH effect, in addition to physical dose sparing. However, the FLASH effect only occurs, when both dose and dose rate meet certain minimum thresholds. This work will develop a simultaneous dose and dose rate optimization (SDDRO) method accounting for both FLASH dose and dose rate constraints during treatment planning for pencil-beam-scanning proton therapy. Methods: SDDRO optimizes the FLASH effect (specific to FLASH-RT) as well as the dose distribution (similar to CONV-RT). The nonlinear dose rate constraint is linearized, and the reformulated optimization problem is efficiently solved via iterative convex relaxation powered by alternating direction method of multipliers. To resolve and quantify the generic tradeoff of FLASH-RT between FLASH and dose optimization, we propose the use of FLASH effective dose based on dose modifying factor (DMF) owing to the FLASH effect. Results: FLASH-RT via transmission beams (TB) (IMPT-TB or SDDRO) and CONV-RT via Bragg peaks (BP) (IMPT-BP) were evaluated for clinical prostate, lung, head-and-neck (HN), and brain cases. Despite the use of TB, which is generally suboptimal to BP for normal tissue sparing, FLASH-RT via SDDRO considerably reduced FLASH effective dose for high-dose OAR adjacent to the target. For example, in the lung SBRT case, the max esophageal dose constraint 27 Gy was only met by SDDRO (24.8 Gy), compared to IMPT-BP (35.3 Gy) or IMPT-TB (36.6 Gy);in the brain SRS case, the brain constraint V12Gy≤15cc was also only met by SDDRO (13.7cc), compared to IMPT-BP (43.9cc) or IMPT-TB (18.4cc). In addition, SDDRO substantially improved the FLASH coverage from IMPT-TB, e.g., an increase from 37.2% to 67.1% for lung, from 39.1% to 58.3% for prostate, from 65.4% to 82.1% for HN, from 50.8% to 73.3% for the brain. Conclusions: Both FLASH dose and dose

关键词： Iterative methods

Macroscopic finite-difference scheme based on the mesoscopic regularized lattice-Boltzmann method

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Physical Review E 2024年第2期109卷 025301-025301页

作者： Xi Liu Ying Chen Zhenhua Chai Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, we develop a macroscopic finite-difference scheme from the mesoscopic regularized lattice Boltzmann (RLB) method to solve the Navier-Stokes equations (NSEs) and convection-diffusion equation (CDE). Unlike the commonly used RLB method based on the evolution of a set of distribution functions, this macroscopic finite-difference scheme is constructed based on the hydrodynamic variables of NSEs (density, momentum, and strain rate tensor) or macroscopic variables of CDE (concentration and flux), and thus shares low memory requirement and high computational efficiency. Based on an accuracy analysis, it is shown that, the same as the mesoscopic RLB method, the macroscopic finite-difference scheme also has a second-order accuracy in space. In addition, we would like to point out that compared with the RLB method and its equivalent macroscopic numerical scheme, the present macroscopic finite-difference scheme is much simpler and more efficient since it is only a two-level system with macroscopic variables. Finally, we perform some simulations of several benchmark problems, and find that the numerical results are not only in agreement with analytical solutions, but also consistent with the theoretical analysis.

关键词： Complex fluids Lattice-Boltzmann methods Navier-Stokes equation

GLOBAL STABILIZABILITY TO TRAJECTORIES FOR THE SCHLÖGL EQUATION IN A SOBOLEV NORM

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arXiv 2022年

作者： Kunisch, Karl Rodrigues, Sérgio S. Johann Radon Institute for Computational and Applied Mathematics ÖAW Altenbergerstrasse 69 Linz4040 Austria Institute for Mathematics and Scientific Computing Heinrichstrasse 36 Graz8010 Austria

The stabilizability to trajectories of the Schlögl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optimality conditions are investigated. These conditions are then used to compute stabilizing receding horizon based controls. Results of numerical simulations are presented comparing their stabilizing performance with that of saturated explicit feedback *** Codes 93C20, 93B52, 93D15, 35K58 Copyright © 2022, The Authors. All rights reserved.

关键词： Trajectories

GENERALIZED TRUNCATED MOMENT PROBLEMS WITH UNBOUNDED SETS

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arXiv 2022年

作者： Huang, Lei Nie, Jiawang Yuan, Ya-Xiang Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Department of Mathematics University of California 9500 Gilman Drive La Jolla San DiegoCA92093 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100049 China

This paper studies generalized truncated moment problems with unbounded sets. First, we study geometric properties of the truncated moment cone and its dual cone of nonnegative polynomials. By the technique of homogenization, we give a convergent hierarchy of Moment-SOS relaxations for approximating these cones. With them, we give a Moment-SOS method for solving generalized truncated moment problems with unbounded sets. Finitely atomic representing measures, or certificates for their nonexistence, can be obtained by the proposed method. Numerical experiments and applications are also given. Copyright © 2022, The Authors. All rights reserved.

关键词： Polynomials

Random ISAC Signals Deserve Dedicated Precoding

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arXiv 2023年

作者： Lu, Shihang Liu, Fan Dong, Fuwang Xiong, Yifeng Xu, Jie Liu, Ya-Feng Jin, Shi Southern University of Science and Technology Shenzhen518055 China The School of Information and Communication Engineering Beijing University of Posts and Telecommunications Beijing100876 China Shenzhen518172 China The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China National Mobile Communications Research Laboratory Southeast University Nanjing210096 China

Radar systems typically employ well-designed deterministic signals for target sensing, while integrated sensing and communications (ISAC) systems have to adopt random signals to convey useful information. This paper analyzes the sensing and ISAC performance relying on random signaling in a multi-antenna system. Towards this end, we define a new sensing performance metric, namely, ergodic linear minimum mean square error (ELMMSE), which characterizes the estimation error averaged over random ISAC signals. Then, we investigate a data-dependent precoding (DDP) scheme to minimize the ELMMSE in sensing-only scenarios, which attains the optimized performance at the cost of high implementation overhead. To reduce the cost, we present an alternative data-independent precoding (DIP) scheme by stochastic gradient projection (SGP). Moreover, we shed light on the optimal structures of both sensing-only DDP and DIP precoders. As a further step, we extend the proposed DDP and DIP approaches to ISAC scenarios, which are solved via a tailored penalty-based alternating optimization algorithm. Our numerical results demonstrate that the proposed DDP and DIP methods achieve substantial performance gains over conventional ISAC signaling schemes that treat the signal sample covariance matrix as deterministic, which proves that random ISAC signals deserve dedicated precoding designs. © 2023, CC BY.

关键词： Numerical methods

POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS II: HIGH DIMENSIONAL CASE

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arXiv 2024年

作者： Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan Institute of Applied Physics and Computational Mathematics Beijing100088 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Sun Yat-sen University Guangdong Zhuhai519082 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, for any odd n and any integer m ≥ 1 with n > 4m, we study the fundamental solution of the higher order Schrödinger equation i∂tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ R, x ∈ Rn, where V is a real-valued C n+21 −2m potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m+V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying n(m−1) |K(t, x, y)| ≤ C(1 + |t|)−(2nm −σ)(1 + |t|− 2nm ) (1 + |t|− 21m |x − y|)− 2m−1 , t , 0, x, y ∈ Rn, where σ = 2 if 0 is an eigenvalue of H, and σ = 0 otherwise. A similar result for smoothing operators H 2αm e−itHPac(H) is also given. The regularity condition V ∈ C n+21 −2m is optimal in the second order case, and it also seems optimal when m > 1. © 2024, CC0.

关键词： Schrodinger equation

PARTIALLY ADJOINT DISCRETIZATIONS OF ADJOINT OPERATORS AND NONCONFORMING FINITE ELEMENT EXTERIOR CALCULUS

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arXiv 2022年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. A theory of nonconforming finite element exterior calculus (NCFEEC) is developed by constructing a systematic family of nonconforming finite element differential forms with certain structures by the theoretical framework and the formal construction of discretizations;by the NCFEEC, together with the classical FEEC, adjoint properties are reconstructed between discretizations of adjoint operators and between discretizations of variational *** Codes 47A05, 47A65, 47N40, 65J10, 65N30 Copyright © 2022, The Authors. All rights reserved.

关键词： Calculations

A hybrid FEM-PINN method for time-dependent partial differential equations

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arXiv 2024年

作者： Feng, Xiaodong Shangguan, Haojiong Tang, Tao Wan, Xiaoliang Zhou, Tao Division of Science and Technology BNU-HKBU United International College Zhuhai519087 China School of Electrical and Computer Engineering Guangzhou Nanfang College Guangzhou510970 China Department of Mathematics Center for Computation and Technology Louisiana State University Baton Rouge70803 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network’s output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments. © 2024, CC BY.

关键词： Partial differential equations